Let \(G = \varinjlim G_n\)  be an inductive limit of compact groups, which is usually not locally compact. When \(G\)  is not locally compact, we cannot utilize standard methods, based on \(C^*\) -algebras, to investigate unitary representations of \(G\) . Nevertheless, Vershik–Kerov's asymptotic representation theory (see [1]}, [2]} and Kerov's thesis [3]}) as well as Olshanski's spherical representation theory (see [4]}, [5]}, [6]}) work well for both the infinite symmetric group \(\mathfrak {S}_\infty = \varinjlim \mathfrak {S}_n\)  (that is locally compact) and the infinite dimensional unitary group \(\mathrm {U}(\infty ) = \varinjlim \mathrm {U}(n)\)  (that is not locally compact) in an almost parallel fashion. This is a bit surprising phenomenon to us, because the natural \(C^*\) -algebras associated with those groups are of different kinds. Actually, the group \(C^*\) -algebra \(C^*(\mathfrak {S}_\infty )\)  is an AF-algebra, while we think, through the analysis of qunatum groups, that the natural \(C^*\) -algebra of \(\mathrm {U}(\infty )\)  should be a \(C^*\) -inductive limit of atomic \(W^*\) -algebras since the associated branching graph is not locally finite.
